cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 36 results. Next

A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

Original entry on oeis.org

1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751, 35090233104309, 140455067207455, 562102681589085, 2249257981411351
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002
As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x) - exp(3x) and a(n) = (4^n - 2*3^n + 2^n)/2. - Paul Barry, May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) is the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x. - Ross La Haye, Jan 10 2008
a(n) also gives the third column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011
a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).

Links

Crossrefs

Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.
Cf. A000079, A000392, A001047, A006516, A032263, A143494, A193685.

Programs

Formula

G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).
a(n-2) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
From Hieronymus Fischer, Jun 25 2007: (Start)
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*4^k.
a(n) = 2^(n+1)*(1+2^(n+2))-3^(n+2). (End)
a(n) = 3*StirlingS2(n+3,4) + StirlingS2(n+3,3). - Ross La Haye, Jan 10 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,2), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011
a(n) = A006516(n+2) - A001047(n+2). - Ross La Haye, Jan 26 2016
a(n) = A006516(n+1) + 3*a(n-1), n>=1, a(0)=1. - Carlos A. Rico A., Jun 22 2019

A051112 Number of monotone Boolean functions of n variables with 4 mincuts. Also Sperner systems with 4 blocks.

Original entry on oeis.org

0, 0, 0, 0, 25, 2020, 82115, 2401910, 58089465, 1245331920, 24625121455, 460316430970, 8266174350005, 144171200793620, 2461016066613195, 41343340015862430, 686274244801356145, 11289648429330100120
Offset: 0

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Author

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

Keywords

References

  • J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, #8, s(n,4).
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean function, Belgrade, 1999, in preparation.

Links

Crossrefs

Cf. A016269, A047707, A051113, A051114, A051115, A051116, A051117, A051118.

Programs

  • Mathematica
    Table[(1/4!)*(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n-22*3^n+6*2^n),{n,0,20}] (* or *) LinearRecurrence[{82,-2970,62700,-856713,7947786,-51019100,226259000,-678011136,1304341632,-1445575680,696729600},{0,0,0,0,25,2020,82115,2401910,58089465,1245331920,24625121455},20] (* Harvey P. Dale, Nov 26 2019 *)
  • PARI
    a(n)=(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n -22*3^n+6*2^n)/24 \\ Charles R Greathouse IV, Mar 14 2012

Formula

a(n) = (1/4!)*(16^n - 12*12^n + 24*10^n + 4*9^n - 18*8^n + 6*7^n - 36*6^n + 36*5^n + 11*4^n - 22*3^n + 6*2^n).
From Michael Somos: (Start)
a(n) = 82*a(n - 1) - 2970*a(n - 2) + 62700*a(n - 3) - 856713*a(n - 4) + 7947786*a(n - 5) - 51019100*a(n - 6) + 226259000*a(n - 7) - 678011136*a(n - 8) + 1304341632*a(n - 9) - 1445575680*a(n - 10) + 696729600*a(n - 11).
G.f.: 5x^4(5-6x-1855x^2+20076x^3-44356x^4-215280x^5+759168x^6) / ((1-3x)(1-4x)(1-5x)(1-6x)(1-2x)(1-7x)(1-8x)(1-9x)(1-10x)(1-12x)(1-16x)). (End)

A051118 Number of monotone Boolean functions of n variables with 10 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1067771, 43506231489, 501425871595264, 2719674203584968630, 9172837864705015158979, 22524989249381408262409893, 44328073635887914351462953684, 74381256243136645820404637874910
Offset: 0

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Author

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

Keywords

References

  • J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
  • V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

Links

Crossrefs

Cf. A016269, A047707, A051112, A051113, A051114, A051115, A051116, A051117.

A084869 Number of 2-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 5, 17, 71, 317, 1415, 6197, 26591, 112157, 466775, 1923077, 7863311, 31972397, 129459335, 522571157, 2104535231, 8460991037, 33972711095, 136277478437, 546270602351, 2188566048077, 8764718254055, 35090241492917
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x = y. - Ross La Haye, Jan 11 2008

Links

Crossrefs

Cf. A000079, A000217, A000392, A016269, A032263, A034472, A047707, A051112-A051118, A084870-A084883.

Programs

  • Mathematica
    Table[2^(2*n-1) - 3^n + 3*2^(n-1), {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2015 *)
  • PARI
    a(n) = 2^(2*n-1)-3^n+3*2^(n-1); \\ Altug Alkan, Sep 12 2017

Formula

a(n) = (1/2!)*(4^n - 2*3^n + 3*2^n).
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 11 2008
G.f.: -(13*x^2-7*x+1) / ((2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Nov 27 2012
a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3). - Vaclav Kotesovec, Oct 30 2015
a(n) = 2^(2n-1) + 2^n + 2^(n-1) - 3^n = A000217(2^n+1) - A034472(n), for n >= 1. - Bob Selcoe, Sep 12 2017

A094033 Number of connected 2-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 3, 18, 75, 270, 903, 2898, 9075, 27990, 85503, 259578, 784875, 2366910, 7125303, 21425058, 64373475, 193317030, 580344303, 1741819338, 5227030875, 15684238350, 47059006503, 141189602418, 423593973075, 1270832250870
Offset: 0

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Author

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

Keywords

Comments

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 10 2008

Links

Crossrefs

Cf. A016269, A047707.
Cf. A051112, A051113, A051114, A051115, A051116, A051117, A051118.
Cf. A094034, A094035, A094036, A094037.

Programs

  • Maple
    [seq(stirling2(n,3)*3,n=0..26)]; # Zerinvary Lajos, Dec 06 2006
  • Mathematica
    Table[3 StirlingS2[n, 3], {n, 0, 26}] (* Michael De Vlieger, Nov 30 2015 *)
  • PARI
    x='x+O('x^50); concat([0,0,0],Vec(serlaplace((exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!))) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = 3 * A000392(n).
E.g.f.: (exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!.
From Colin Barker, Mar 31 2012: (Start)
a(n) = (3^n-3*2^n+3)/2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
G.f.: 3*x^3/((1-x)*(1-2*x)*(1-3*x)). (End)

A094037 Number of connected 6-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 1, 1345, 738741, 185165477, 29458046177, 3541242666045, 354515664467077, 31326419674855789, 2535191648955942273, 192567615994193565125, 13962461827318220986133, 978010022290154153870661
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

Keywords

Crossrefs

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

Formula

E.g.f.: (exp(63*x) - 30*exp(47*x) + 120*exp(39*x) + 60*exp(35*x) + 60*exp(33*x) - 18*exp(32*x) - 339*exp(31*x) - 720*exp(29*x) + 810*exp(27*x) + 120*exp(26*x) + 480*exp(25*x) + 480*exp(24*x) - 600*exp(23*x) - 720*exp(22*x) - 240*exp(21*x) - 900*exp(20*x) + 1740*exp(19*x) + 615*exp(18*x) + 180*exp(17*x) + 435*exp(16*x) - 1445*exp(15*x) - 3270*exp(14*x) + 1710*exp(13*x) + 4620*exp(12*x) - 3360*exp(11*x) - 3210*exp(10*x) + 3360*exp(9*x) + 6810*exp(8*x) - 12465*exp(7*x) + 5985*exp(6*x) + 7110*exp(5*x) - 18555*exp(4*x) + 17884*exp(3*x) - 8352*exp(2*x) + 1764*exp(x) - 120)/6!.

A084883 Number of (k,m,n)-multiantichains of multisets with k=3 and m=6.

Original entry on oeis.org

1, 3, 64, 8022, 6822072, 14068794534, 26314469636622, 37310026340520678, 42667193588371160460, 42169580808988409450310, 37803058273249518925923210, 31733179110752959606870643334
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118, A084869-A084882.

Formula

a(n) = (1/6!)*(729^n - 30*486^n + 120*378^n + 60*324^n + 60*294^n - 360*279^n - 12*276^n - 720*252^n + 45*243^n + 90*234^n + 720*231^n + 120*216^n + 720*210^n - 240*205^n + 360*196^n - 720*189^n - 180*187^n + 720*186^n - 720*176^n + 120*168^n - 720*167^n + 360*165^n - 900*162^n - 720*157^n + 180*156^n + 720*148^n - 240*145^n + 720*138^n + 30*134^n - 240*129^n + 2700*126^n - 360*120^n + 180*111^n + 900*108^n - 20*102^n + 450*98^n - 5400*93^n - 5400*84^n + 685*81^n + 1350*78^n + 5400*77^n + 5400*70^n - 5400*63^n + 900*56^n - 8220*54^n + 16440*42^n + 2740*36^n - 16440*31^n + 4275*27^n + 4110*26^n - 25650*18^n + 25650*14^n + 10474*9^n - 20948*6^n + 7560*3^n).

A051113 Number of monotone Boolean functions of n variables with 5 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 6, 2146, 304752, 25400564, 1557306954, 78817977462, 3513106214484, 143429796694888, 5501383287745422, 201652447559180618, 7148287976359243896, 247151326758617289372, 8386495692534098616210, 280574309728711561269214, 9286566498536162168164188
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

Keywords

References

  • J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
  • V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean function, Belgrade, 1999, in preparation.

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118.

Formula

a(n) = 1/5! * (32^n-20 * 24^n+ 60 * 20^n+ 20 * 18^n+ 10 * 17^n-110 * 16^n-120 * 15^n+ 150 * 14^n+ 120 * 13^n-240 * 12^n+ 20 * 11^n+ 240 * 10^n+ 40 * 9^n-205 * 8^n+ 60 * 7^n-210 * 6^n+ 210 * 5^n+ 50 * 4^n-100 * 3^n+ 24 * 2^n).
G.f.: -2*x^4*(140561100029952000*x^15 -73258140662784000*x^14 -8396658614522880*x^13 +15284070825850368*x^12 -4918391338514880*x^11 +748203166795520*x^10 -45197506544400*x^9 -3280961201664*x^8 +887950976060*x^7 -80597007540*x^6 +3942400065*x^5 -98697251*x^4 +532770*x^3 +26970*x^2 -335*x -3) / ((2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)*(11*x -1)*(12*x -1)*(13*x -1)*(14*x -1)*(15*x -1)*(16*x -1)*(17*x -1)*(18*x -1)*(20*x -1)*(24*x -1)*(32*x -1)). - Colin Barker, Jul 14 2013

Extensions

More terms from Colin Barker, Jul 14 2013

A051114 Number of monotone Boolean functions of n variables with 6 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1380, 759457, 192504214, 31169837405, 3827970163920, 392135190780649, 35468973527445018, 2937270598777421269, 228156280366446932500, 16904255174464832812001, 1208995011493806361868862, 84197134590686932418878093, 5746616155270206518199693720
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

Keywords

References

  • J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
  • V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

Links

Crossrefs

Cf. A016269, A047707, A051112, A051113, A051115, A051116, A051117, A051118.

Formula

a(n) = (1/6!)*(64^n-30 * 48^n+ 120 * 40^n+ 60 * 36^n+ 60 * 34^n-12 * 33^n-345 * 32^n-720 * 30^n+ 810 * 28^n+ 120 * 27^n+ 480 * 26^n+ 360 * 25^n-480 * 24^n-720 * 23^n-240 * 22^n-540 * 21^n+ 1380 * 20^n+ 750 * 19^n+ 60 * 18^n-210 * 17^n-1535 * 16^n-1820 * 15^n+ 2250 * 14^n+ 1800 * 13^n-2820 * 12^n+ 300 * 11^n+ 2040 * 10^n+ 340 * 9^n-1815 * 8^n+ 510 * 7^n-1350 * 6^n+ 1350 * 5^n+ 274 * 4^n-548 * 3^n+ 120 * 2^n).

Extensions

More terms from Colin Barker, Nov 26 2014

A056005 Number of 3-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 3 labeled nodes and n hyperedges.

Original entry on oeis.org

0, 0, 0, 2, 19, 90, 302, 820, 1926, 4068, 7920, 14454, 25025, 41470, 66222, 102440, 154156, 226440, 325584, 459306, 636975, 869858, 1171390, 1557468, 2046770, 2661100, 3425760, 4369950, 5527197, 6935814, 8639390, 10687312, 13135320, 16046096, 19489888
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jul 24 2000

Keywords

Comments

T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes have a hyperedge containing one but not the other node.

Examples

			There are 19 3-element ordered antichains on an unlabeled 4-element set: ({4},{3},{2}), ({4},{3},{1,2}), ({4},{2,3},{1}), ({4},{2,3},{1,3}), ({3,4},{2},{1}), ({3,4},{2},{1,4}), ({3,4},{2,4},{2,3}), ({3,4},{2,4},{1}), ({3,4},{2,4},{1,4}), ({3,4},{2,4},{1,3}), ({3,4},{2,4},{1,2}), ({3,4},{2,4},{1,2,3}), ({3,4},{1,2},{2,4}), ({3,4},{1,2,4},{2,3}), ({3,4},{1,2,4},{1,2,3}), ({2,3,4},{1,4},{1,3}), ({2,3,4},{1,4},{1,2,3}), ({2,3,4},{1,3,4},{1,2}), ({2,3,4},{1,3,4},{1,2,4}).

Links

Crossrefs

Cf. A047707 for 3-element (unordered) antichains on a labeled n-element set.
Cf. A056069, A056070, A056071, A056073, A056163.

Programs

  • Magma
    [n*(n-2)*(n-1)*(n+1)*(n^3 + 30*n^2 + 131*n - 270)/5040: n in [0..30]]; // G. C. Greubel, Nov 22 2017
  • Mathematica
    Table[Binomial[n+7,7]-6Binomial[n+5,5]+6Binomial[n+4,4]+3Binomial[n+3,3]- 6Binomial[n+2,2]+ 2Binomial[n+1,1],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,0,0,2,19,90,302,820},40] (* Harvey P. Dale, Jul 27 2011 *)
  • PARI
    x='x+O('x^50); concat([0,0,0], Vec(x^3*(2+3*x-6*x^2+2*x^3)/(1-x)^8)) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = C(n+7, 7) - 6*C(n+5, 5) + 6*C(n+4, 4) + 3*C(n+3, 3) - 6*C(n+2, 2) + 2*C(n+1, 1).
a(n) = n*(n-2)*(n-1)*(n+1)*(n^3 + 30*n^2 + 131*n - 270)/5040.
G.f.: 1/(1-x)^8 - 6/(1-x)^6 + 6/(1-x)^5 + 3/(1-x)^4 - 6/(1-x)^3 + 2/(1-x)^2.
G.f.: x^3*(2 + 3*x - 6*x^2 + 2*x^3)/(1-x)^8.
Recurrence: a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
Generally, recurrence for the number of m-element ordered antichains on an unlabeled n-element set is a(m, n) = C(2^m, 1)*a(m, n - 1) - C(2^m, 2)*a(m, n - 2) + C(2^m, 3)*a(m, n - 3) + ... + ( - 1)^(k - 1)*C(2^m, k)*a(m, n - k) + ... - a(m, n - 2^m).
a(n) = A000580(n+7) - 6*A000389(n+5) + 6*A000332(n+4) + 3*A000292(n+1) - 6*A000217(n+1) + 2*A000027(n+1). - R. J. Mathar, Nov 16 2007

Extensions

More terms from Harvey P. Dale, Jul 27 2011
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